Chapman–Ferraro problem

Remarks

The surface is now called the magnetopause, and the currents on it are the Chapman–Ferraro currents. It is a free-boundary problem: the shape of the surface determines the field, and the field determines the shape. A first estimate uses the subsolar point. If the particles reflect specularly, the stream exerts a pressure $2\rho v^2$ there. Let $B_0$ be the equatorial surface field and $R_E$ the Earth's radius. Assume the boundary currents double the dipole field $B_0 (R_E/r)^3$ just inside the surface. Then the balance $2\rho v^2 = (2 B_0 (R_E/r)^3)^2/(2\mu_0)$ gives the stand-off distance $$\frac{r}{R_E} = \left( \frac{B_0^2}{\mu_0 \rho v^2} \right)^{1/6}.$$ For typical solar-wind values this gives about ten Earth radii, in line with observations. The full shape of the surface has been computed numerically. Modern models also include the magnetic field carried by the solar wind and the reconnection that lets some of it enter. Chapman and Ferraro posed the problem in their work on magnetic storms from 1930 on, before the continuous solar wind was known.